Chapter 11, Problem 1RE

Solution

a)

To calculate: The value of the limit $\underset{x\to 1^{-1}}{\lim }f\left(x\right)$ using the given graph.

The value of the limit of the given function at $x=1$ is 2.

Given information:

The graph is given below.

  

The left-hand limit of a function at a point is defined as the approaching value of the function from the left side at that particular point.

From the provided graph, it can be observed that the function approaches to 2 from left side of $x=1$ . This implies that,

  $\begin{array}{c}\text{LHL}=\underset{x\to 1^{-}}{\lim }f\left(x\right)\\ =2\end{array}$

Thus, the left-hand limit of the given function at $x=1$ is 2.

b)

To calculate: The value of the limit $\underset{x\to 1}{\lim }f\left(x\right)$ using the given graph.

The limit of the given function at $x=1$ dos not exist.

Given information:

The graph is given below.

  

The left-hand limit of a function at a point is defined as the approaching value of the function from the left side at that particular point.

From the provided graph, it can be observed that the function approaches to 2 from left side of $x=1$ . This implies that,

  $\begin{array}{c}\text{LHL}=\underset{x\to 1^{-}}{\lim }f\left(x\right)\\ =2\end{array}$

The right-hand limit of a function at a point is defined as the approaching value of the function from the right side at that particular point.

From the provided graph, it can be observed that the function approaches to 1 from right side of $x=1$ . This implies that,

  $\begin{array}{c}\text{RHL}=\underset{x\to 1^{+}}{\lim }f\left(x\right)\\ =1\end{array}$

It is known that the limit of a function at a point exist if left-hand limit and right limit at that point are equal. From the above calculations, it can be concluded that $\text{LHL}\ne \text{RHL}$ .

Thus, the limit of the given function at $x=1$ does not exist.